One of the things that turned me off OSM back in the beginning (a few years ago; I can't quite remember when) was its seeming focus on making maps for CARS. I don't like cars! I cycle, or walk, and don't have a driver's license; I want to contribute to maps for walkers and cyclists, not for those silly GPS things that everyone's suctioning to their windscreens these days. (Rhubarb rhubarb.)

Thankfully, I have since realised that OSM is about MAPS, not cars, and that I can contribute whatever parts of the map that I feel are useful. So, I find myself focusing on paths for pedestrians, and because I live in the 'burbs, most of those paths are a) foopaths next to residential streets, and b) paths of desire (oh, such a delightful term!) in those rare bits of non-suburb'd land.

The question is: should one trace these paths individually? I say yes, because the information cannot be captured with tags on a road. But some people say no. Hmm....

I would say if the path is immediately adjacent to the road, separated only by a kerb (eg a sidewalk / pavement), then adding the footway=both/left/right tag to the road is best.

But if the path is more separated from the road (eg with grass in between), then worth mapping it separately. As this means you can show where it actually is, and whether its parallel to the road etc, plus you can tag it with the surface / width / access.
Though if doing this make sure you map where it connects to the road, to help with routing.

If you map the sidewalk next to a road, the routing will just not work. It won't know that you can cross the road at any point. As a result it will send you on some strange detour to a node that happens to be connected across a road just to get to the other side. And if the footway isn't well connected, you may find some segments totally unroutable.

A footway is an object for it's own, and should also represented as such. Car routing can esay filter out footways by tag. Adding the footways to the road is like adding the next post office address to every postbox next to it (done by Deutsche Post AG in Germany) instead to mark as as amenity=post_office.

I only add urban footways and cycleways when they actually go somewhere useful, like a footway cut-through between roads which could potentially save a pedestrian user hundreds of meters on their journey.